static void
test_random(const size_t N, const gsl_rng *r, const int compress)
{
const gsl_splinalg_itersolve_type *T = gsl_splinalg_itersolve_gmres;
const double tol = 1.0e-8;
int status;
gsl_spmatrix *A = create_random_sparse(N, N, 0.3, r);
gsl_spmatrix *B;
gsl_vector *b = gsl_vector_alloc(N);
gsl_vector *x = gsl_vector_calloc(N);
/* these random matrices require all N iterations to converge */
gsl_splinalg_itersolve *w = gsl_splinalg_itersolve_alloc(T, N, N);
const char *desc = gsl_splinalg_itersolve_name(w);
create_random_vector(b, r);
if (compress)
B = gsl_spmatrix_compcol(A);
else
B = A;
status = gsl_splinalg_itersolve_iterate(B, b, tol, x, w);
gsl_test(status, "%s random status s=%d N=%zu", desc, status, N);
/* check that the residual satisfies ||r|| <= tol*||b|| */
{
gsl_vector *res = gsl_vector_alloc(N);
double normr, normb;
gsl_vector_memcpy(res, b);
gsl_spblas_dgemv(CblasNoTrans, -1.0, A, x, 1.0, res);
normr = gsl_blas_dnrm2(res);
normb = gsl_blas_dnrm2(b);
status = (normr <= tol*normb) != 1;
gsl_test(status, "%s random residual N=%zu normr=%.12e normb=%.12e",
desc, N, normr, normb);
gsl_vector_free(res);
}
gsl_spmatrix_free(A);
gsl_vector_free(b);
gsl_vector_free(x);
gsl_splinalg_itersolve_free(w);
if (compress)
gsl_spmatrix_free(B);
} /* test_random() */
/*
test_poisson()
Solve u''(x) = -pi^2 sin(pi*x), u(x) = sin(pi*x)
epsrel is the relative error threshold with the exact solution
*/
static void
test_poisson(const size_t N, const double epsrel, const int compress)
{
const gsl_splinalg_itersolve_type *T = gsl_splinalg_itersolve_gmres;
const size_t n = N - 2; /* subtract 2 to exclude boundaries */
const double h = 1.0 / (N - 1.0); /* grid spacing */
const double tol = 1.0e-9;
const size_t max_iter = 10;
size_t iter = 0;
gsl_spmatrix *A = gsl_spmatrix_alloc(n ,n); /* triplet format */
gsl_spmatrix *B;
gsl_vector *b = gsl_vector_alloc(n); /* right hand side vector */
gsl_vector *u = gsl_vector_calloc(n); /* solution vector, u0 = 0 */
gsl_splinalg_itersolve *w = gsl_splinalg_itersolve_alloc(T, n, 0);
const char *desc = gsl_splinalg_itersolve_name(w);
size_t i;
int status;
/* construct the sparse matrix for the finite difference equation */
/* first row of matrix */
gsl_spmatrix_set(A, 0, 0, -2.0);
gsl_spmatrix_set(A, 0, 1, 1.0);
/* loop over interior grid points */
for (i = 1; i < n - 1; ++i)
{
gsl_spmatrix_set(A, i, i + 1, 1.0);
gsl_spmatrix_set(A, i, i, -2.0);
gsl_spmatrix_set(A, i, i - 1, 1.0);
}
/* last row of matrix */
gsl_spmatrix_set(A, n - 1, n - 1, -2.0);
gsl_spmatrix_set(A, n - 1, n - 2, 1.0);
/* scale by h^2 */
gsl_spmatrix_scale(A, 1.0 / (h * h));
/* construct right hand side vector */
for (i = 0; i < n; ++i)
{
double xi = (i + 1) * h;
double bi = -M_PI * M_PI * sin(M_PI * xi);
gsl_vector_set(b, i, bi);
}
if (compress)
B = gsl_spmatrix_compcol(A);
else
B = A;
/* solve the system */
do
{
status = gsl_splinalg_itersolve_iterate(B, b, tol, u, w);
}
while (status == GSL_CONTINUE && ++iter < max_iter);
gsl_test(status, "%s poisson status s=%d N=%zu", desc, status, N);
/* check solution against analytic */
for (i = 0; i < n; ++i)
{
double xi = (i + 1) * h;
double u_gsl = gsl_vector_get(u, i);
double u_exact = sin(M_PI * xi);
gsl_test_rel(u_gsl, u_exact, epsrel, "%s poisson N=%zu i=%zu",
desc, N, i);
}
/* check that the residual satisfies ||r|| <= tol*||b|| */
{
gsl_vector *r = gsl_vector_alloc(n);
double normr, normb;
gsl_vector_memcpy(r, b);
gsl_spblas_dgemv(CblasNoTrans, -1.0, A, u, 1.0, r);
normr = gsl_blas_dnrm2(r);
normb = gsl_blas_dnrm2(b);
status = (normr <= tol*normb) != 1;
gsl_test(status, "%s poisson residual N=%zu normr=%.12e normb=%.12e",
desc, N, normr, normb);
gsl_vector_free(r);
}
gsl_splinalg_itersolve_free(w);
gsl_spmatrix_free(A);
gsl_vector_free(b);
gsl_vector_free(u);
if (compress)
gsl_spmatrix_free(B);
} /* test_poisson() */
static void
test_toeplitz(const size_t N, const double a, const double b,
const double c)
{
int status;
const double tol = 1.0e-10;
const size_t max_iter = 10;
const gsl_splinalg_itersolve_type *T = gsl_splinalg_itersolve_gmres;
const char *desc;
gsl_spmatrix *A;
gsl_vector *rhs, *x;
gsl_splinalg_itersolve *w;
size_t i, iter = 0;
if (N <= 1)
return;
A = gsl_spmatrix_alloc(N ,N);
rhs = gsl_vector_alloc(N);
x = gsl_vector_calloc(N);
w = gsl_splinalg_itersolve_alloc(T, N, 0);
desc = gsl_splinalg_itersolve_name(w);
/* first row */
gsl_spmatrix_set(A, 0, 0, b);
gsl_spmatrix_set(A, 0, 1, c);
/* interior rows */
for (i = 1; i < N - 1; ++i)
{
gsl_spmatrix_set(A, i, i - 1, a);
gsl_spmatrix_set(A, i, i, b);
gsl_spmatrix_set(A, i, i + 1, c);
}
/* last row */
gsl_spmatrix_set(A, N - 1, N - 2, a);
gsl_spmatrix_set(A, N - 1, N - 1, b);
/* set rhs vector */
gsl_vector_set_all(rhs, 1.0);
/* solve the system */
do
{
status = gsl_splinalg_itersolve_iterate(A, rhs, tol, x, w);
}
while (status == GSL_CONTINUE && ++iter < max_iter);
gsl_test(status, "%s toeplitz status s=%d N=%zu a=%f b=%f c=%f",
desc, status, N, a, b, c);
/* check that the residual satisfies ||r|| <= tol*||b|| */
{
gsl_vector *r = gsl_vector_alloc(N);
double normr, normb;
gsl_vector_memcpy(r, rhs);
gsl_spblas_dgemv(CblasNoTrans, -1.0, A, x, 1.0, r);
normr = gsl_blas_dnrm2(r);
normb = gsl_blas_dnrm2(rhs);
status = (normr <= tol*normb) != 1;
gsl_test(status, "%s toeplitz residual N=%zu a=%f b=%f c=%f normr=%.12e normb=%.12e",
desc, N, a, b, c, normr, normb);
gsl_vector_free(r);
}
gsl_vector_free(x);
gsl_vector_free(rhs);
gsl_spmatrix_free(A);
gsl_splinalg_itersolve_free(w);
} /* test_toeplitz() */
int
main()
{
const size_t N = 100; /* number of grid points */
const size_t n = N - 2; /* subtract 2 to exclude boundaries */
const double h = 1.0 / (N - 1.0); /* grid spacing */
gsl_spmatrix *A = gsl_spmatrix_alloc(n ,n); /* triplet format */
gsl_spmatrix *C; /* compressed format */
gsl_vector *f = gsl_vector_alloc(n); /* right hand side vector */
gsl_vector *u = gsl_vector_alloc(n); /* solution vector */
size_t i;
/* construct the sparse matrix for the finite difference equation */
/* construct first row */
gsl_spmatrix_set(A, 0, 0, -2.0);
gsl_spmatrix_set(A, 0, 1, 1.0);
/* construct rows [1:n-2] */
for (i = 1; i < n - 1; ++i)
{
gsl_spmatrix_set(A, i, i + 1, 1.0);
gsl_spmatrix_set(A, i, i, -2.0);
gsl_spmatrix_set(A, i, i - 1, 1.0);
}
/* construct last row */
gsl_spmatrix_set(A, n - 1, n - 1, -2.0);
gsl_spmatrix_set(A, n - 1, n - 2, 1.0);
/* scale by h^2 */
gsl_spmatrix_scale(A, 1.0 / (h * h));
/* construct right hand side vector */
for (i = 0; i < n; ++i)
{
double xi = (i + 1) * h;
double fi = -M_PI * M_PI * sin(M_PI * xi);
gsl_vector_set(f, i, fi);
}
/* convert to compressed column format */
C = gsl_spmatrix_ccs(A);
/* now solve the system with the GMRES iterative solver */
{
const double tol = 1.0e-6; /* solution relative tolerance */
const size_t max_iter = 10; /* maximum iterations */
const gsl_splinalg_itersolve_type *T = gsl_splinalg_itersolve_gmres;
gsl_splinalg_itersolve *work =
gsl_splinalg_itersolve_alloc(T, n, 0);
size_t iter = 0;
double residual;
int status;
/* initial guess u = 0 */
gsl_vector_set_zero(u);
/* solve the system A u = f */
do
{
status = gsl_splinalg_itersolve_iterate(C, f, tol, u, work);
/* print out residual norm ||A*u - f|| */
residual = gsl_splinalg_itersolve_normr(work);
fprintf(stderr, "iter "F_ZU" residual = %.12e\n", iter, residual);
if (status == GSL_SUCCESS)
fprintf(stderr, "Converged\n");
}
while (status == GSL_CONTINUE && ++iter < max_iter);
/* output solution */
for (i = 0; i < n; ++i)
{
double xi = (i + 1) * h;
double u_exact = sin(M_PI * xi);
double u_gsl = gsl_vector_get(u, i);
printf("%f %.12e %.12e\n", xi, u_gsl, u_exact);
}
gsl_splinalg_itersolve_free(work);
}
gsl_spmatrix_free(A);
gsl_spmatrix_free(C);
gsl_vector_free(f);
gsl_vector_free(u);
return 0;
} /* main() */
